Double-refraction optical transmission filters can consist of a double-refraction crystal plate arranged between two parallel linear polarizers. In such an arrangement the filter possesses the characteristic feature that the first polarizer in the direction of propagation of the light determines the polarization state with which the light of a wide-spectrum beam impinges upon the crystal. Because of the double refraction, which can be assumed generally without limitation to be a linear double refraction, the light beam is split into two beams polarized orthogonally and propagating at different velocities in the crystal. The light beam which issues from the crystal, and which results from the superposition of the two partial beams, exhibits generally elliptical polarization. The second polarizer, on the output side, is transparent only to the light beam component which is parallel to its plane of polarization and its transmission varies sinusoidally with the phase delay applied to one partial beam relative to the other because of the crystal double refraction. Since--without consideration of the variation in the effective index of refraction determined by the crystal material dispersion--said delay is approximately in inverse proportion to the wavelength of the incident light, the transmission is sinusoidally dependent on the inverse wavelength.
For a given wavelength .lambda. of the light injected into the double-refraction crystal with a defined linear polarization state, the transmission is maximum in such a filter when the relative phase shift is 2.pi. or an integral multiple .rho. of 2.pi., determined by the difference in the two optical distances travelled by the two orthogonal polarization states. This can be written as the equation: ##EQU1## where p is the order in which the filter is operated at given wavelength .lambda..
This condition is met also for proximate wavelengths .lambda..sub.1 =.lambda.+.DELTA..lambda..sub.1 and .lambda..sub.2 =.lambda.-.DELTA..lambda..sub.2 when: ##EQU2## The wavelength intervals .DELTA..lambda..sub.1,2 for the transmission maxima of the filter transmission maximum for wavelength .lambda. is then given by the equation: ##EQU3##
It is apparent that a given minimum distance .DELTA..lambda. between the transmission maxima of such a filter can be maintained only when the order in which the filter is operated is not larger than a maximum p.sub.max obtained from equation (3) for wavelength interval .DELTA..lambda..sub.1 of the "longer" wavelength when this equation is solved according to p: ##EQU4## This means that such a transmission filter, which for a wavelength .lambda. of 480 nm is to have an interval .DELTA..lambda. of about 20 nm between transmission maxima, must be operated with a maximum order of p=25.
From equation (1) it follows directly that a calcite plate (.DELTA.n=about 0.16) with plane-parallel surfaces mounted as a double-refraction element in a transmission filter operated at a wavelength .lambda.=500 nm with order 25, must present a thickness d of about 0.081 mm. For a satisfactory filter operation accuracy of about 1/100 of .lambda., thickness d.sub..lambda. within which a phase delay 2.pi. occurs between the two orthogonal polarization states, is required, i.e. 30 nm or 1/17 of the wavelength. Therefore, the crystal plate must be processed with high accuracy, which naturally entails very high manufacturing costs for the filter. This is true also when, instead of a calcite plate, a quartz plate is used as a double-refraction element in which the difference .DELTA.n between the indices of refraction applicable to the two orthogonal polarization states is about 0.01. In this case the .lambda. thickness is about 50,000 nm, and the tolerance acceptable for the quartz-plate thickness is about 500 nm, which is thus on the order of magnitude of the wavelength of the light to be filtered.
This disadvantage in the difficult and expensive production of such crystal plates applies particularly to filter arrangements in which a plurality of crystal plates are mounted in succession as a stack along the beam path. The first multilayer arrangement of this type was a filter proposed by Lyot (B. Lyot, Ann. Astrophys. 1944:7(1), 2). The Lyot filter comprises, for example, N plates stacked successively in the direction of light propagation, each plate being used with double the thickness of the preceding plates. Each plate is mounted with the polarizers crossed at a right angle. The optical axes of the double-refraction delay plates extend at 45.degree. to the planes of polarization defined by the polarizers. The resulting transmission presents very definite transmission maxima with a stop-band which is determined by the plates of least thickness, and whose bandwidth decreases as the number of plates increases. Weakly marked secondary maxima also exist between the primary maxima. The transmission bandwidth obtainable with a Lyot filter comprising up to 10 plates is typically 5 to 0.5 A.
Similar narrow bandwidths are obtained with the multilayer double-refraction transmission filter suggested by Solc (I. Solc, Czechoslov. Cosopis pro Fysiku 1953:3, 336; 1954:4, 607, 609; 1955:5, 114). In a structure roughly equivalent to that of a Lyot filter with N plates, the Solc filter comprises, for example, m plates of equal thickness d equal to the thickness of the thinnest plates of the Lyot filter. The entire stack of plates is arranged between only two polarizers. The optical axes of the individual crystal plates are parallel to the plate surfaces and perpendicular to the direction of light propagation. In a first embodiment of the Solc filter the optical axes of the individual crystal plates are shifted fanlike by an angle EQU .omega..sub.j =(.xi./2)+(j-1).xi. (5)
with EQU .xi.=.pi./2m (6)
where m is the number of plates of equal thickness. The polarizers are parallel.
In a second equivalent embodiment the directions of the optical axes alternate successively at an angle EQU .omega..sub.j =(-1).sup.j+1 (.xi./2) (7)
where equation (6) applies to .xi.. The polarizers between which the stack of plates is arranged are crossed.
A further description of other properties of the Lyot filter and the Solc fan and folded filters can be found in a comparative description by John W. Evans (Journal of the Optical Society of America, Vol. 48, No. 3, March 1958, pp. 142ff.).
It is apparent that, because of the multiplicity of necessary crystal plates which must be processed with the above-cited accuracy, and the need to maintain exactly the orientation of the crystal-plate optical axes, the production of both the Lyot filter and the Solc filter is extremely complicated and expensive. Therefore, it is very difficult to tune such filters.